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Abstract We study the low-rank phase retrieval problem, where our goal is to recover a $$d_1\times d_2$$ low-rank matrix from a series of phaseless linear measurements. This is a fourth-order inverse problem, as we are trying to recover factors of a matrix that have been observed, indirectly, through some quadratic measurements. We propose a solution to this problem using the recently introduced technique of anchored regression. This approach uses two different types of convex relaxations: we replace the quadratic equality constraints for the phaseless measurements by a search over a polytope and enforce the rank constraint through nuclear norm regularization. The result is a convex program in the space of $$d_1 \times d_2$$ matrices. We analyze two specific scenarios. In the first, the target matrix is rank-$$1$$, and the observations are structured to correspond to a phaseless blind deconvolution. In the second, the target matrix has general rank, and we observe the magnitudes of the inner products against a series of independent Gaussian random matrices. In each of these problems, we show that anchored regression returns an accurate estimate from a near-optimal number of measurements given that we have access to an anchor matrix of sufficient quality. We also show how to create such an anchor in the phaseless blind deconvolution problem from an optimal number of measurements and present a partial result in this direction for the general rank problem.
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Simultaneous Blind Deconvolution and Phase Retrieval with Tensor Iterative Hard Thresholding
Blind deconvolution and phase retrieval are both fundamental problems with a growing interest in signal processing and communications. In this work, we consider the task of simultaneous blind deconvolution and phase retrieval. We show that this non-linear problem can be reformulated as a low-rank tensor recovery problem and propose an algorithm named TIHT-BDPR to recover the unknown parameters. We include a series of numerical simulations to illustrate the effectiveness of our proposed algorithm.
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- Award ID(s):
- 1704204
- PAR ID:
- 10099568
- Date Published:
- Journal Name:
- ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- Page Range / eLocation ID:
- 2977 to 2981
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICASSP.2019.8683575
